Three - dimensional Geometry

• Direction cosines and direction ratios of a line joining two points

• Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines

• Cartesian and vector equation of a plane

• Angle between −

  • Two lines

  • Two planes

  • A line and a plane

• Distance of a point from a plane

SUMMARY

1. Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.

2. If l, m, n are the direction cosines of a line, then l^2 + m^2 + n^2 = 1.

3. Direction cosines of a line joining two points P(x1 , y1 , z1 ) and Q(x2 , y2 , z2 )

4. Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.

5. If l, m, n are the direction cosines and a, b, c are the direction ratios of a line.

6. Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.

7. Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.

8. If l1 , m1 , n1 and l 2 , m2 , n2 are the direction cosines of two lines; and q is the acute angle between the two lines; then cosθ = | l1 l2 + m1 m2 + n1 n2 |

9. Shortest distance between two skew lines is the line segment perpendicular to both the lines.

10. Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n is lx + my + nz = d.

11. Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x1 , y1 , z1 ) is A (x – x1 ) + B (y – y1 ) + C (z – z1 ) = 0